If $$ (\sqrt{\frac{7}{5}})^{3x-y} = \frac{875}{2401} $$ and $$ (\frac{4a}{b})^{6x-y} = (\frac{2a}{b})^{y-6x} $$, for all non-zero real values of $$ a $$ and $$ b $$, then the value of $$ x + y $$ is:

Question

Accepted Answer

### Step 1: Simplify the First Equation  
The first equation is:  
$$
\left(\sqrt{\frac{7}{5}}ight)^{3x-y} = \frac{875}{2401}.
$$  
Rewriting the bases:  
$$
\left(\frac{7}{5}ight)^{\frac{3x-y}{2}} = \frac{125}{343}.
$$  
Expressing $$ \frac{125}{343} $$ as $$ \left(\frac{7}{5}ight)^{-3} $$:  
$$
\left(\frac{7}{5}ight)^{\frac{3x-y}{2}} = \left(\frac{7}{5}ight)^{-3}.
$$  
Equating the powers:  
$$
\frac{3x-y}{2} = -3.
$$  
Multiply through by 2 to simplify:  
$$
3x - y = -6. 	ag{1}
$$

### Step 2: Simplify the Second Equation  
The second equation is:  
$$
\left(\frac{4a}{b}ight)^{6x-y} = \left(\frac{2a}{b}ight)^{y-6x}.
$$  
Rewriting the bases, let $$ R = \frac{2a}{b} $$:  
$$
R^{6x-y} = R^{y-6x}.
$$  
For these powers to be equal, the exponents must satisfy:  
$$
6x - y = y - 6x.
$$  
Simplify:  
$$
12x = 2y.
$$  
Divide through by 2:  
$$
y = 6x. 	ag{2}
$$

### Step 3: Solve for $$ x $$ and $$ y $$ Using Both Equations  
Substitute $$ y = 6x $$ from Equation (2) into Equation (1):  
$$
3x - 6x = -6.
$$  
Simplify:  
$$
-3x = -6 \implies x = 2.
$$

Now substitute $$ x = 2 $$ into $$ y = 6x $$:  
$$
y = 6(2) = 12.
$$

### Step 4: Calculate $$ x + y $$:  
$$
x + y = 2 + 12 = 14.
$$

Thus, the value of $$ x + y $$ is **14**.